Luckily, perhaps sensing the confusion over these moves, John Tortorella sought to clarify at least one, candidly offering this explanation of the decision to waive Avery. From the New York Daily News:

"I don't want to jam up Sean here - I think we have better players than Sean Avery, plain and simple," Tortorella said in a small gathering with reporters on the second floor of the team's hotel. "I can dodge it ten different ways, without trying to run Sean over. I thought he had a good camp. But I think with the makeup of our team, and some of the people we've added, and some of the youth we've added as far as depth put Sean in this spot."

And then, in case perhaps he hadn't made himself perfectly clear, Torts returned to the point:

"The players that are here are better than Sean Avery," Tortorella reiterated. " Maybe not in the role or a couple instances Sean could help us in, but they're more versatile in a lot of different areas.

"I'm trying to do this the right way, because I do not wanna keep on shoveling dirt over Sean Avery, but we have better players than Sean Avery, right now on the hockey club."

I'm still not entirely sure what he's trying to say.

Tortorella sounds like a guy who's mind has been made up for awhile. One wonders why the Rangers even brought Avery to Europe with them. Think they were trying to give him a head start on looking for a new job?

According to Blue Seat Blogs, the notion that three teams can split Avery's contract is a misconception. Only two teams can split a contract:

[...] if the Rangers put Avery on regular waivers and is claimed by a third team, then team 3 assumes the Rangers' responsibilities. The Stars and team 3 now split Avery's salary & cap hit.

If Avery isn't claimed and clears waivers, but the Rangers later recall him from the minors and another team claims him on the way back up, team 3 splits the cost with the Rangers. Dallas is now off the hook.

And section 50.9 of the CBA backs this up:

To the extent the Player does require Waivers to be Loaned to a minor league affiliate, he cannot be Loaned or recalled without first clearing regular Waivers, and then cannot be Recalled to the NHL parent Club during the same League Year without also clearing a new Re-Entry Waiver procedure, pursuant to which the Player can be claimed by another NHL Club for fifty (50) percent of the contract's remaining amounts to be paid, with the balance to be paid by and charged to the waiving NHL Club (both amounts to be counted against each Club's Upper Limit, Actual Club Salary and Averaged Club Salary, and counted against the Players' Share); [...]

Again, the CBA only allows for two teams to split a contract fifty/fifty.

UPDATE: According to the NHL, this is how it would break down for Avery.

If Avery were claimed on re-entry waivers, his salary would be paid by and count against three teams' caps with this breakdown: Dallas 50 percent, Rangers 25 percent, New Team 25 percent.